Charge Density Wave

Summary codes used to calculate electronic susceptibility (for phonon softening) and spectral function (for electronic gap opening) in my CDW research.
Updating without a fixed schedule.
The detailed underlying physics can be found in my publications, the related physical quantities are briefly shown below.

Phonon instability (q-dependent)

Fermi Surface Nesting (FSN)

• Static Lindhard susceptibility

$${\chi'}_\mathbf{q}=\sum_{\mathbf{k}}\frac{f(\varepsilon_\mathbf{k})-f(\varepsilon_{\mathbf{k}+\mathbf{q}})}{\varepsilon_{\mathbf{k}+\mathbf{q}}-\varepsilon_\mathbf{k}}$$

• Nesting function

$${\chi''}_\mathbf{k}=\sum_{\mathbf{q}}\delta\left(\varepsilon_\mathbf{k}\right)\delta\left(\varepsilon_{\mathbf{k}+\mathbf{q}}\right)$$

Electron-Phonon Coupling (EPC)

• Moment-dependent electron-phonon coupling

$${\bar{g}}_\mathbf{q}=\sum_{\mathbf{k}}\left|g_{\mathbf{k},\mathbf{k}+\mathbf{q}}\right|^2$$

FSN + EPC

• Generalized static electronic susceptibility

$$\chi_\mathbf{q}=\sum_{\mathbf{k}}{\left|g_{\mathbf{k},\mathbf{k}+\mathbf{q}}\right|^2\frac{f(\varepsilon_\mathbf{k})-f(\varepsilon_{\mathbf{k}+\mathbf{q}})}{\varepsilon_{\mathbf{k}+\mathbf{q}}-\varepsilon_\mathbf{k}}},$$

Scattered electrons (k-dependent)

Transfer sum over $\mathbf{q}$ to sum over $\mathbf{k}$ is from my institution, which should correspond to the scattered electrons (electron instability).

Elastic scattered electrons

$${\chi'}_\mathbf{k}=\sum_{\mathbf{q}}\frac{f\left(\varepsilon_\mathbf{k}\right)-f\left(\varepsilon_{\mathbf{k}+\mathbf{q}}\right)}{\varepsilon_{\mathbf{k}+\mathbf{q}}-\varepsilon_\mathbf{k}}$$ $${\chi''}_\mathbf{k}=\sum_{\mathbf{q}}\delta\left(\varepsilon_\mathbf{k}\right)\delta\left(\varepsilon_{\mathbf{k}+\mathbf{q}}\right)$$

Inelastic scattered electrons

$$\chi_\mathbf{k}=\sum_{\mathbf{q}}{\left|g_{\mathbf{k},\mathbf{k}+\mathbf{q}}\right|^2\frac{f(\varepsilon_\mathbf{k})-f(\varepsilon_{\mathbf{k}+\mathbf{q}})}{\varepsilon_{\mathbf{k}+\mathbf{q}}-\varepsilon_\mathbf{k}}}$$

Electron instability

The spectral function is calculated to show the CDW gaps

• Mean-field Hamiltonian

$$H_{mf}=\sum_{\mathbf{k}}{\varepsilon_\mathbf{k}c_\mathbf{k}^\dagger c_\mathbf{k}+\sum_{\mathbf{k},\mathbf{Q}}{2g_{\mathbf{k},\mathbf{k}+\mathbf{Q}}\Delta_\mathbf{Q}c_\mathbf{k}^\dagger c_{\mathbf{k}+\mathbf{Q}}}}+h.c..$$

• Retarded Green’s function

$$G_R(\omega) = (\omega + i \eta - H_{\text{mf}} )^{-1}.$$

Publications

If you use these CDW codes in your work, please consider citing:

Z. Wang, J. Zhou, K. P. Loh, and Y. P. Feng, Controllable phase transitions between multiple charge density waves in monolayer 1T-VSe₂ via charge doping. Appl. Phys. Lett. 119, 163101 (2021)
Z. Wang, C. Chen, J. Mo, J. Zhou, K. P. Loh, and Y. P. Feng, Decisive role of electron-phonon coupling for phonon and electron instabilities in transition metal dichalcogenides. Phys. Rev. Research 5, 013218 (2023)
Z. Wang, J-Y. You, C. Chen, J. Mo, J. He, L. Zhang, J. Zhou, K. P. Loh, and Y. P. Feng, Interplay of the charge density wave transition with topological and superconducting properties. Nanoscale Horizons 8, 1395 (2023)